A revised version of “Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors” has been released on arXiv today.
Among other improvements, the revised version incorporates additional remarks on the connection to the existing literature on stable distributions in Banach spaces, and generalises the results of the previous version of the paper to quasi-Banach spaces, which are like complete normed vector spaces in every respect except that the triangle inequality only holds in the weakened form

This semester, Winter Semester 2016–2017, I will be teaching the third-semester course Stochastik I for mathematics bachelors' degree students at the Free University of Berlin.
Exercise sheets, announcements, etc. for this course will all be posted on this page, as well as on the official FU Berlin webpages such as KVV.

Abstract. Probabilistic numerics is an emerging field which constructs probability measures to capture uncertainty arising from the discretisation which is often necessary to solve complex problems numerically.
We explore probabilistic numerical methods for Partial differential equations (PDEs).
We phrase solution of PDEs as a statistical inference problem, and construct probability measures which quantify the epistemic uncertainty in the solution resulting from the discretisation [1].

We analyse these probability measures in the context of Bayesian inverse problems, parameter inference problems whose dynamics are often constrained by a system of PDEs. Sampling from parameter posteriors in such problems often involves replacing an exact likelihood with an approximate one, in which a numerical approximation is substituted for the true solution of the PDE.
Such approximations have been shown to produce biased and overconfident posteriors when error in the forward solver is not tightly controlled. We show how the uncertainty from a probabilistic forward solver can be propagated into the parameter posteriors, thus permitting the use of coarser discretisations while still producing valid statistical inferences.

Han Cheng Lie and I have just uploaded a preprint of our latest paper, on Cameron–Martin-type theorems for sequences of Cauchy-distributed random variables, to the arXiv.
Inspired by questions of prior robustness left unanswered in this earlier paper on α-stable Banach space priors, this paper addresses the basic probabilistic question:
when is an infinite-dimensional Cauchy distribution, e.g. on sequence space, mutually absolutely continuous with its image under a translation?
In the Gaussian case, the celebrated Cameron–Martin theorem says that this equivalence of measures holds if a weighted \(\ell^{2}\) norm (the Cameron–Martin norm) of the translation vector is finite.
We show that, in the Cauchy case, the same weighted version of the translation vector needs to lie in the sequence space \(\ell^{1} \cap \ell \log \ell\).
More precisely, if the Cauchy distribution on the nth term of the sequence has width parameter \(\gamma_{n} > 0\), and the translation vector is the sequence \(\varepsilon = (\varepsilon_n)_{n = 1}^{\infty}\), then a sufficient condition for mutual absolute continuity is that